Topological properties of mappings from the Sobolev classes with summable Jacobian (Q1594209)

Let \(n\geq 2\), \(G\subset \mathbb R ^n\). For a mapping \(f:G\to \mathbb R ^n\), \(f=(f_1, \dots{}, f_n)\) assume that all coordinate functions \(f_i\) belong to the Sobolev space \(W^1_{p,\text{loc}}(G)\); thus the formal Jacobian matrix \(Df(x)\) and its Jacobian \(J(x,f)=\det Df(x)\) are defined for almost all points of the domain \(G\). The author considers mappings \(f\in W^1_{q,\text{loc}}(G)\) that satisfy the following conditions: M1. \(q\geq 1\) for \(n=2\) and \(q>n-1\) for \(n\geq 3\); M2. \(J(x,f)\geq 0\); M3. \(J(x,f)\in L_{1,\text{loc}}(G)\); M4. The condition \(J(x,f)=0\) almost everywhere on the set \(A\subset G\), \(|A|>0\) implies \(Df(x)=0\) almost everywhere on \(A\); M5. The mapping \(f:G\to\mathbb R^n\) is continuous. The paper contains a survey of various topological properties such as monotonicity, classical differentiability, preserving orientation, etc., for mappings as above. Proofs are only sketched or bibliographical references are indicated.